This is one of the more remote parts of China.  There is a very sizeable Muslim minority in the city and it is one of the poorer parts of the country.  Since you are from Australia, I should tell you that there is really never, ever torrid summer weather -- summer in Xining probably resembles spring or late winter in Oz.  Air quality in the city is considered second worst in the country after Beijing and some days it is worse than Beijing.  For a small city that size, that's a remarkable negative achievement.  Han Chinese predominate but there is also a noticeable Tibetan minority. The major train line runs to Lanzhou and from there there are connections to the rest of the country.  The regional airport is not large but there are flight to the major Chinese cities.  There is a large well-known mosque in the city.

By Chinese standards, I would consider the city to be moderately safe.  Be careful of the railway station -- there are hordes of pickpockets there.  Ditto the bus station.  At night travel by taxi, if you can afford it, and make sure that there are no other passengers in the cab when you get in and that the driver picks up no other passengers.

Re private teaching, I have no idea on that issue except to say that since this is a relatively impoverished area, what you might earn from private teaching would not be much.

Bring a warm fleece coat.  Stay safe.